Unsteady motion of nearly spherical particles in viscous fluids: a second-order asymptotic theory

The motion of small non-spherical particles is often studied using the unsteady Stokes equations. Zhang & Stone (J. Fluid Mech., vol. 367, 1998, pp. 329–358) reported an asymptotic treatment for nearly spherical particles, to first order in particle non-sphericity, i.e. $O(\epsilon )$, where $\epsilon$ quantifies the shape deviation from a sphere. Importantly, key physical phenomena are absent at $O(\epsilon )$, including (1) coupling between the torque experienced by the particle and its linear translation, (2) coupling between the force the particle experiences and its rotation and (3) the effect of non-sphericity on the orientation averages of these forces and torques. We present an explicit asymptotic theory to second order in particle non-sphericity, i.e. $O(\epsilon ^2)$, for the force and torque acting on a particle in a general unsteady Stokes flow. The derived analytical formulae apply to particles of arbitrary shape, providing the leading-order asymptotic theory for the three above-mentioned phenomena. The theory is demonstrated for several example nearly spherical particles including a spheroid, a ‘pear-shaped’ particle and a simple model for a SARS-CoV-2 virion. This includes formulae for force and torque as a function of particle orientation and their corresponding orientation averages. Our study reveals that the orientation-averaged forces and torques experienced by a nearly spherical particle cannot be generally represented by a perfect sphere. The reported formulae are validated using finite-amplitude three-dimensional direct numerical simulations of the Navier–Stokes equations. A Mathematica notebook is also provided, facilitating implementation of the theory for particle shapes of the user's choosing.